I tend to treat words fairly literally and try to be precise, so if I say ‘I’m sorry,’ it’s because I am sorry. If I want to sympathize, I can say other things. (‘That’s awful.’ ‘Those bastards…’ etc.) On occasions where I have not felt sorry because the other person has reacted emotionally to something , I tend to ask questions. Calmly giving people a chance to feel heard can often help.
In most cases, though, I try to move people toward solutions-oriented thinking to prevent spiralling. Asking questions lets them put the problem into words, helping them switch from a defensive emotional stance to an open brainstorming stance. Giving people a goal can make them feel a sense of progress, giving them distance from the problem and possibly netting a better final result than even might have happened without the inciting event.